Some basic electronics and truth tables Some material on truth tables can be found in Chapter 3 of Digital Principles (Tokheim)

Logic  Digital Electronics In Logic, one refers to Logical statements (propositions which can be true or false). What a computer scientist would represent by a Boolean variable. In Electronics, one refers to inputs which will be high or low.

Boola Boola! The expression (Booleans) and the rules for combining them (Boolean algebra) are named after George Boole ( ), a British mathematician.

Boolean operators AND: when two or more Boolean expressions are ANDed, both must be true for the combination to be true. OR: when two or more Boolean expressions are ORed, if either one or the other or both are true, then the combination is true. NOT: takes one Boolean expression and yields the opposite of it, true  false and vice versa.

Representations of Standard Boolean Operators Boolean algebra expression Gate symbol NOT AA´A´ A AND BAB A OR BA+B A XOR B ABAB A NOR B(A+B)´ A NAND B(AB)´

Our Notation NOT is represented by a prime or an apostrophe. A’ means NOT A OR is represented by a plus sign. A + B means A OR B AND is represented by placing the two variables next to one another. AB means A AND B The notation is like multiplication in regular algebra since if A and B are 1’s or 0’s the only product that gives 1 is when A and B are both 1.

Other Notations means NOT A A means NOT A AB means A OR B A&B means A AND B Tokheim uses the overbar notation for NOT, but we will use the prime notation because it is easier to type.

Other vocabulary We will tend to refer to A and B as “inputs.” (Electronics) Another term for them is “Boolean variables.” (Programming) Still another term for them is “propositions.” (Logic) And yet another term for them is “predicates.” (Logic and grammar)

(AB)’  A’B’ ABAB(AB)’ ABA’B’A’B’ Note that the output is different

A Truth Table A Truth table lists all possible inputs, that is, all possible values for the propositions. For a given numbers of inputs, this is always the same. Then it lists the output for each possible combination of inputs. This varies from situation to situation.

The true one Traditionally we take a 1 to represent true and a 0 to represent false In addition, we will usually interpret a high voltage as a true and a low voltage as a false

Generating Inputs The truth-table inputs consist of all the possible combinations of 0’s and 1’s for that number of inputs. One way to generate the inputs for is to count in binary. For two inputs, the combinations are 00, 01, 10 and 11 (binary for 0, 1, 2 and 3). For three inputs, the combinations are 000, 001, 010, 011, 100, 101, 110 and 111 (binary for 0, 1, 2, 3, 4, 5, 6 and 7). For n inputs there are 2 n combinations (rows in the truth table).

Expressing truth tables Every truth table can be expressed in terms of the basic Boolean operators AND, OR and NOT operators. The circuits corresponding to those truth tables can be build using AND, OR and NOT gates. The input in each line of a truth table can be expressed in terms of AND’s and NOT’s.

ABA’B’ ABA’B ABAB’ ABAB Note that these expressions have the property that their truth table output has only 1 row with a 1.

In a sense, each line has an expression Input AInput B Expression 00 (NOT A) AND (NOT B) A´B´ 01 (NOT A) AND B A´B 10 A AND (NOT B) AB´ 11 A AND B ABAB

It’s true; it’s true The following steps will allow you to generate an expression for the output of any truth table. Take the true (1) outputs Write the expressions for that input line (as shown on the previous slide) Then feed all of those expressions into an OR gate Sometimes we have multiple outputs (e.g. bit addition had a sum output and a carry output). Then each output is treated separately.

Example: Majority Rules ABCMajority If two or more of the three inputs are high, then the output is high.

Row Expressions ABCRow expressions 000A’B’C’ 001A’B’C 010A’BC’ 011A’BC 100AB’C’ 101AB’C 110ABC’ 111ABC The highlighted rows correspond to the high outputs.

Majority rules A´BC + AB´C + ABC´ + ABC NOTs ANDs OR

Electronics Workbench Where the logic gates are

EWB gates chips Be careful above the NAND chip is the NOT gate.

NOT Gate and NOT Chip

Gates and Chips EWB gates focus purely on the logic (the inputs and outputs). EWB chips, while simulated, are closer to a real-world device Chips need power: VCC must be connected to high (5 volts) and GND must be connected to ground. Another complication is that the NOT chip houses four NOT gates.

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Connectors: allow up to four wires to meet

Resistors:

Component Properties: Right click

Value and units Value of resistance Kilo-Ohm: Unit of resistance

Switch

Battery and ground Ground Battery

Voltmeter, ammeter, 7-segment display(s)

Venn Diagram A Venn diagram is a pictorial representation of a truth table. Venn diagrams come from set theory. The correspondence between set theory and logic is that either one belongs to a set or one does not, so set theory and logic go together.

Venn (Cont.) Belongs to set  True Does not belong to set  False

Overlapping sets A true, but B false B true, but A false A false and B false A and B true

Ohm’s Law V = I R, where V is voltage: the amount of energy per charge. I is current: the rate at which charge flows, e.g. how much charge goes by in a second. R is resistance: the “difficulty” a charge encounters as moves through a part of a circuit.

Circuit A circuit is a closed path along which charges flow. If there is not a closed path that allows that the charge can get back to where it started (without retracing its steps), the circuit is said to be “open” or “broken.” The path doesn’t have to be unique; there may be more than one path.

An analogy A charge leaving a battery is like you starting the day after a good night’s rest; you are full of energy. Being the kind of person you are, you will expend all of your energy and collapse utterly exhausted into bed at the end of the day; the charge uses up all of its energy in traversing a circuit.

Analogy (cont.) You look ahead to the tasks of the day and divide your energy accordingly, the more difficult the task the more of your energy it requires (resistors in series). The tasks are resistors, so more energy (voltage) is used up working through the more difficult tasks (higher resistances). The higher the resistance, the greater the voltage drop (energy used up) across it.

One charge among many You are just one charge among many. If the task at hand is very difficult (the resistance is high), not many will do it (the current is low); V=IR, if R is big, I must be small. If the task is easy, everyone rushes to do it. V=IR, if R is small, I will be large.

More energetic If we had more energy, more of us would attempt a given task. V=IR, if V is bigger, I is bigger. If we are all tired out, few of us will perform even the most basic task. V=IR, if V is small, I will be small.

Given the choice Given the choice between a difficult task and an easy task, most will choose the easier task. If there is more than one path, most take the “path of least resistance” (resistors in parallel).